I agree that a 50Hz motor supplied with 60Hz will run faster.
I don't agree with your HP reasoning. The HP needed is defined by the load, the motor has to provide it. With something like a fan the mechanical power varies with the square of the speed, so doubling the speed requires 4 times the mechanical power & will shift a lot more air.
So, if the motor is running faster then the load is likely to demand more mechanical power & this may well overload the motor.
The current in the windings has two components - the first is a reactive component that creates the magnetic field. I agree that this will be less at 60Hz. The other component is that current which is responsible for the 'work'. This will be directly proportional to the mechanical load - since the mechanical load is likely to be greater (its being spun faster) then this component of the current will be bigger.
